Convex Metric Space - Convex Metric Spaces and Convex Sets

Convex Metric Spaces and Convex Sets

As mentioned in the examples section, closed subsets of Euclidean spaces are convex metric spaces if and only if they are convex sets. It is then natural to think of convex metric spaces as generalizing the notion of convexity beyond Euclidean spaces, with usual linear segments replaced by metric segments.

It is important to note, however, that metric convexity defined this way does not have one of the most important properties of Euclidean convex sets, that being that the intersection of two convex sets is convex. Indeed, as mentioned in the examples section, a circle, with the distance between two points measured along the shortest arc connecting them, is a (complete) convex metric space. Yet, if and are two points on a circle diametrally opposite to each other, there exist two metric segments connecting them (the two arcs into which these points split the circle), and those two arcs are metrically convex, but their intersection is the set which is not metrically convex.